Find the equation in standard form for the hyperbola that satisfies the given conditions: transverse axis has endpoints (5,3) and (-7, 3) and conjugate axis has a length of 10.
I found the distance of the transverse axis to be 12. For the formula, I have a=6 b=5. I just need help finding h and k.
Thanks!
(h,k) is the center. The center is the midway point between the two vertices: (5,3) and (-7,3). Using the midpoint formula, ((5-7)/2,(3-3)/2), you get h=-1 and k=0. hope this helps you Katelyn.
To find the values of h and k for the equation of the hyperbola in the standard form, you can use the midpoint formula.
Step 1: Find the midpoint of the transverse axis
To find the midpoint, you can use the formula:
Midpoint = ( (x1 + x2) / 2, (y1 + y2) / 2 )
Using the endpoints of the transverse axis:
Midpoint = ( (5 + (-7)) / 2, (3 + 3) / 2 )
= ( (-2) / 2, 6 / 2 )
= ( -1, 3 )
So, the midpoint of the transverse axis is (-1, 3).
Step 2: Identify the values of h and k
In the standard form of the equation of a hyperbola, the center is represented as (h, k).
Therefore, h = -1 and k = 3.
Now you can write the equation of the hyperbola in standard form using the given values of a, b, h, and k.
Recall that the standard form of the equation for a hyperbola with center (h, k) is:
( (x - h)^2 / a^2 ) - ( (y - k)^2 / b^2 ) = 1
Substituting the given values, the equation becomes:
( (x - (-1))^2 / 6^2 ) - ( (y - 3)^2 / 5^2 ) = 1
Simplifying the equation, you get:
( (x + 1)^2 / 36 ) - ( (y - 3)^2 / 25 ) = 1
Thus, the equation of the hyperbola in standard form is:
( (x + 1)^2 / 36 ) - ( (y - 3)^2 / 25 ) = 1.
To find the equation of the hyperbola in standard form, you need to determine the values of h and k, which represent the coordinates of the center. Here's how you can find them:
1. Start by finding the midpoint of the transverse axis. The midpoint is the average of the x-coordinates of the endpoints of the transverse axis and the average of the y-coordinates of the endpoints. In this case, the x-coordinates of the endpoints are 5 and -7, so the average is (5 + (-7))/2 = -1/2. The y-coordinates of the endpoints are both 3, so the average is (3+3)/2 = 3.
Therefore, the coordinates of the center (h, k) are (-1/2, 3).
2. Now that you have the values of a=6, b=5, and the center (h, k) = (-1/2, 3), you can plug these values into the standard form equation for a hyperbola with a transverse axis parallel to the x-axis:
(x - h)^2/a^2 - (y - k)^2/b^2 = 1
Substituting the given values, the equation becomes:
(x - (-1/2))^2/6^2 - (y - 3)^2/5^2 = 1
Simplifying further:
(x + 1/2)^2/36 - (y - 3)^2/25 = 1
So, the equation of the hyperbola in standard form is:
(x + 1/2)^2/36 - (y - 3)^2/25 = 1.